Noncrossing partitions and Milnor fibers

Abstract

For a finite real reflection group W we use non-crossing partitions of type W to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated W -discriminant ∆W and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of ∆W .

On the homology of the noncrossing partition lattice and the Milnor fibre

“…Definition 6.1. [BFW18]The NCP model F of the Milnor fibre F Q " Q ´1p1q is the pn ´1qdimensional finite simplicial complex whose k-simplices are of the form pm, w, e ă w 1 ă ¨¨¨ă w k q, 0 ď m ă n ´ℓT pw k q, w P W, where e ă w 1 ă ¨¨¨ă w k is a chain of L. Each k-simplex has k `1 maximal faces: pm, w, e ă w 1 ă ¨¨¨ă x w i ă ¨¨¨ă w k q for 1 ď i ď k and the remaining one pm `ℓT pw 1 q, ww 1 , e ă w ´1 1 w 2 ă ¨¨¨ă w ´1 1 w k q. Note that by the condition on the integer m, the element w k in the k-simplex pm, w, e ă w 1 ă ¨¨¨ă w k q should be strictly less than the Coxeter element γ of L. The Coxeter group W acts on the k-simplex by left multiplication on w. The generator of the monodromy group of F Q sends the k-simplex to pm ´1, w, e ă w 1 ă ¨¨¨ă w k q if 0 ă m ă n ´ℓT pw k q and p´1q k pℓ T pw 1 q ´1, ww 1 , e ă w ´1 1 w 2 ă ¨¨¨ă w ´1 1 w k ă w ´1 1 γq if m " 0.…”

“…Brady, Falk and Watt gave a finite simplicial complex which has the homotopy type of the Milnor fibre F Q0 [BFW18]. This simplicial complex is described in terms of the NCP lattice and hence is called the NCP model of F Q0 .…”

“…Similarly, they introduced the NCP model of F P which is a CW complex. Using a filtration by subcomplexes of the NCP model, they obtained a chain complex which computes the integral homology of F P , with the chain groups being the top homology groups of some rank-selected subposets of L. However, an explicit description of those top homology groups and boundary maps was in absence [BFW18,Theorem 6.4], and the NCP model of F Q0 is cumbersome for computational purposes.…”

“…As another application, we obtain two chain complexes which compute H ˚pM ; Zq and H ˚pM {W ; Zq in terms of the algebra B in Theorem 7.2 and Theorem 7.5, respectively. This is again based on the NCP models of the hyperplane complement M and its quotient M {W [BFW18]. It is well known that the (co)homology of M is torsion free [Bri73] and the cohomology ring H ˚pM ; Zq is isomorphic to the Orlik-Solomon algebra [OS80].…”

“…The NCP model of F P is also introduced in [BFW18]. It is defined to be the pn ´1qdimensional CW-complex F W whose k-cells are of the form pm, e ă w 1 ă ¨¨¨ă w k q with 0 ď m ă n ´ℓT pw k q, where e ă w 1 ă ¨¨¨ă w k is a chain of L. Note that F W is not a simplicial complex, and each cell of F W can be obtained from a simplex of F by removing the group element w in the middle of the triple (cf.…”